Once upon a time I looked for a formula which incorporated the PEG < 1: the stock is a good buy
>Okay, so you wanted to use the PEG Ratio to decide when to buy and when to ...
Speed is measured in kilometres per hour
>That's like: (km / hours)*(hours) = km, right?
Now suppose you saw a formula like
>The right side is (dollars per share per year)*(dollars per share) so ...
>That's like Distance = Speed * Time, in Japan or the U.S or ...
>Or e = m c It's curious, don't you think? In Science one never says:
"The force is 15 Newtons" or "The mass is 20 kilograms".
In Finance, one often leaves out the dimensions, eh? >Like "The P/E Ratio is 32" and "The PEG Ratio is 0.60"
Exactly! Just think of how scary it'd be if one contemplated buying a stock with a P/E Ratio of 200 years!
>That one's already in my portfolio! Pay attention. >Okay, what about the dimensions of "percentage"?
>So the dimensions would cancel out, eh?
Okay, here are some examples of financial objects and their dimensions: - P/E Ratio = (Dollars per Share) / (Dollars per Share per Year) = Years
- PEG = (P/E) / (percentage growth per Year) = Years
^{2} - Market Capitalization = (Shares) * (Dollars per Share) = Dollars
- Volatility (or Standard Deviation) of annual returns = percentage change in stock Price, per Year = 1/Years
- Book Value = Dollars per Share
- Earnings = Dollars per Share per Year
- Stock Return = percentage change in Price per Year = 1/Years
- Sharpe Ratio = (average Return - risk-free Return) / (Standard Deviation of Returns) =
*dimensionless* - DOW Index = Average of 30 stock Prices = Dollars.per Share
- S&P 500 Index = SUM of (Market Caps) = Dollars
Yeah. Nice, eh? >But don't you think that PEG, as Years ^{2}, is strange?
Yes, very strange. Whereas one can attach some significance to P/E as Years, namely the number of years that it'd take current earnings to equal the stock price, I have no idea how to interpret PEG as Years ^{2}.
>Do you know anything else measured in Years ^{2} ?
No. Well ... maybe, yes. If a ball is dropped from a building, the distance it falls in T seconds is (about): [*] distance = (1/2)acceleration*T ^{2}
So we could write: (distance)/(acceleration) as seconds ^{2}.
>I meant financial stuff. No. >By the way, did you say that the DOW is the >For A = $12 and B = $8 then d = (4+8)/(12+8) = 0.60
[1] Portfolio = P _{0} e^{rT}
where P _{0} = initial portfolio,
r = annual return (as a percentage per year) and T = number of years. Now, it just so happens that e Note that here we're adding 1 to x to x Let's consider the formula:
Here we're adding 1 and r so r should be dimensionless (since "1" has no dimensions).
Let's change [2] into [1]: - Assume we calculate the gain N times per year with a fractional change of r/N each time period.
- Then we'd have, in one year, N fractional increases, each r/N.
- Each dollar would grow to (1+r/N) each time we do the compounding (which is N times per year).
- That'd mean that our portfolio (originally Po) would have grown to Po (1+r/N)
^{NT}after T years (meaning after NT time periods). - Letting N become
*infinite*and noting that (1+r/N)^{N}becomes e^{r}(as N becomes*infinite*), we get: Portfolio = Po e^{rT}which is just formula [1].
>But didn't you say that, in [1], r had the dimension of 1/Time?
Yes, but we're going to fix that! Look again at P My point is just this:
That means that both sides of an equation should have the same dimensions ... and formulas should never involve the addition of quantities with different dimensions.
Let's consider some financial entity that has the dimensions of, say, (Dollars per Share)*(Years).
Many ... not all?
No, of course not. The value of your portfolio will be in some convenient currency units: dollars or yen or francs or ... >So what financial entities should be dimensionless? Well ... uh, if you use a financial entity to decide when to buy or sell, maybe that entity should be dimensionless.
>Dimensionless, eh? That's important? Who knows. I just have a hunch that they'd be useful ... like the Sharpe Ratio which is a ratio - hence a comparison - of two like quantities. After all, if you say "The P/E is low", I'd say "Low compared to what?".
Anyway, we'll see. Give me a few months to check this out, okay? >You said the units were Dollars and Years and ... Well, what I really mean is Currency Units and Time Units and Share Units.
We're looking for something, namely "d", with the dimensions Years ^{2}, so PEG / d would be dimensionless.
We could consider:
If the market caps were M _{1}, M_{2}, ... M_{n} our portfolio would be worth
M_{1}+M_{2}+ ... +M_{n} dollars.
Suppose the total Earnings for each of the n companies were E_{1}, E_{2}, ... E_{n} dollars per year.
The P/E Ratio would be (M _{1}+M_{2}+ ... +M_{n}) / (E_{1}+E_{2}+ ... +E_{n}) years.
>P/E Ratio? Yes, we'll need it to calculate a PEG. Note that P/E has the dimensions of Years ... or Time units. Suppose that we owned a total of N shares (or Share units).
The value of our portfolio would then be A = (M _{1}+M_{2}+ ... +M_{n}) / N dollars per share.
>You mean Currency units per Share units?
Yes. Of course. Further, the Earnings would be B = (E _{1}+E_{2}+ ... +E_{n}) / N dollars per share per year.
Note that, for our basket of stocks,
Okay, back to our PEG(index). Our earnings growth
>Remind me. Why are we doing this? We want to introduce a normalized/standardized/dimensionless PEG Ratio for a stock, so we divide the standard PEG Ratio by the PEG Ratio for some appropriate index, like the DOW or the S&P500 or the Wiltshire5000 or ... >Okay, okay. So do it.
If our stock were a large cap growth stock, we'd probably want the index to consist of large cap growth stocks. Indeed, our stock might very well be included in the index. Suppose it is. In fact, suppose it's the first stock. The
Suppose there are n _{1} outstanding shares of our stock and the market cap is M_{1} dollars.
Then the price is M _{1}/n_{1} dollars per share and the earnings are E_{1}/n_{1}
dollars per share per year.
The P/E Ratio is then (M _{1}/n_{1}) / (E_{1}/n_{1}) = M_{1}/E_{1} years.
The Earnings Growth Rate is r _{1} per year so we get
PEG(stock) = M_{1}/(r_{1}E_{1}).
>Isn't that just [C], with n = 1? Uh, yeah, I guess it is Anyway, forging ahead we get: _{1}/(r_{1}E_{1})] /
[
(M_{1}+M_{2}+ ... +M_{n}) /
( r_{1}E_{1}+ r_{2}E_{2}+ ... + r_{n}E_{n})
]
This can also be written like so: _{1}/(M_{1}+M_{2}+ ... +M_{n})]
/
[r_{1}E_{1}/( r_{1}E_{1}+ r_{2}E_{2}+ ... + r_{n}E_{n})]
which we see is just (market cap as a percentage of the total index market cap) /(earnings growth as a percentage of the total index growth) >Example?
>And if that's the smallest gPEG, that'd be the best buy? I didn't say that! >Then what are you suggesting?
Me? Suggest? I just write tutorials and spreadsheets, do charts, provide food for thought ... >Yeah, right. Are you deep in thought? >No! And do you realize that you don't have a single chart?
I've already told you ... I have no idea. >But what about bpp's suggestion, at NFB?
Oh, yeah. That's neat. Consider the reciprocal, 1/PEG: - 1/PEG = (Earnings Growth Rate)/(Price/Earnings) = r/(P/E) = rE/P.
- r, the Growth Rate, is measured in percentage per year.
- E, the annual earnings, is in dollars per share per year.
- So rE (in dollars per share, per year per year, ) gives the rate at which the annual dollars of earnings grow, per share.
- Dividing by the stock price P (in dollars per share) gives a neat comparison of the rate at which the
earnings grow to the actual stock price ... and it's measured in 1/years
^{2}.
If the Earnings Per Share, E, is at E = $0.50 and grows to $0.60 in a year, that's a 20% growth per year so r = 0.20 and since E = $0.50 then rE = 0.20*0.50 = 0.10 so if earnings continue to grow in this way we'd expect a growth of $0.10 dollars per year, every year.
>Per share. Yes, per share. But that increase per year, every year, is like that ball dropping from a tall
building. Its speed increases 32 feet per second, every second.
Now, that $0.10 dollars per year, every year: Is it good or bad?
>For a $2 stock, that's good: PEG = (P/E)/r = (2/0.5)/20 = 0.2. For a $100 stock, that's bad: PEG = 10. Exactly! So we compare rE to the stock price by dividing by P. Neat, eh? >And that means ... what? It means that: - If earnings have an annual increase of rE dollars per share,
*every*year - then the increases would be rE, 2rE, 3rE, ... nrE after 1, 2, 3, ... n years, and
- nrE is measured in (dollars per share per year),
- so, after n years, annual earnings will have increased to E+nrE = E(1+nr) dollars per share per year,
- and that's the same as P provided E(1+nr) = P or n = [(P/E)-1]/r years = (P/E)/r - 1/r = 100PEG - 1/r.
Because our r is a fraction, like r = 0.20, whereas PEG uses a percentage, like 20%. For example, if PEG = 0.8 and r = 0.25 (that's a 25% annual earnings growth) then n = 100PEG - 1/r = 80 - 4 = 76 years. >What! 76 years! I'll be dead by then! Yes, me too, but this is fiction. We're assuming constant earnings growth and ... >Dimensional inconsistency! Dimensional inconsistency! Huh? >That equation: n = 00PEG - 1/r ? Since n is in years and PEG is in years ^{2} and 1/r is in ...
Uh oh, yes. The left side is years, the right side is years ^{2} + years. Uh ... let's do it differently:
- We're n years into the future now ... and the annual earnings are E(1+nr) dollars per share.
- We'd like the P/E ratio (n years in the future) to be, say ,
**k**based solely upon earnings growth ... not on stock price. - The stock price is then
*still*P dollars per share, the EPS is then E(1+nr) dollars per share per year. - We want P/E(after n years) = P/[E(1+nr)] =
**k**years ... so we can solve for n = 100PEG/**k**- 1/r years.
Let's do that example again: If PEG = 0.8 and r = 0.25 (that's a 25% annual earnings growth) then n = 100PEG/ k - 1/r = 80/k - 4 years.
>And if I ... uh, if you wanted a P/E of k = 1 then you'd have to wait 76 years! No wonder it was so long!
Yes, but waiting for a P/E of 8 is only 80/8 - 4 = 6 years. >And how long until we've got a P/E of k = 40?
This is fiction, remember? However, have you found the dimensional "1" in that inconsistent equation: n = 100PEG - 1/r ?
>Yes, it's n = 100PEG/ 1 - 1/r. That's with k = 1.
Ain't this dimensional stuff fun? >Not really.
Now let's look for some dimensionless quantity, some number we can attach to a stock, to tell us if it's a good buy.
We'd like to have a We'll give this number a name:
>I assume the g stands for ...
For golden, as in the golden NUMber.
>Sure. Our point is, even for a low-PEG stock, if the volatility is large we may be in trouble. >Especially if Volatility = Risk, eh? Please don't equate Volatility and Risk. Just think of Volatility as being bad. After all, if a stock has an average return of, say, R, then the annualized return (and that's MUCH more important than the average return) ... the annualized return is very nearly R - Volatility ^{2}/2 so it decreases if the Volatility increases.
>So it becomes riskier! Not necessarily ... if R increases dramatically as well. Anyway, don't start that risk = volatility argument again $#@%!*? Okay, now we calculate a >And buy the one with the smallest !
gNUMI didn't say that! Remember, I don't give investment advice. I write tutorials, do charts ... >Yeah, yeah. I'll tell you something. I don't like that name: gNUM. What would you suggest? >Remember that movie with Peter Sellers and his "Birdie Num Num"? What would you suggest?
>I don't suggest. I nit-pick, make trouble, find your errors ... Pay attention!
Okay, here's something else of interest.
- We buy all the stock. That's the market cap $M.
- Unfortunately, we also inherit the company's debt. That's $D.
- Aah, but we acquire the company's cash and any marketable securities they hold. That's $C.
- Total
__Net__Cost: M+D-C called theor*Enterprise Value***EV**dollars.
Okay, now we look at Earnings: - We find the company's total annual Earnings before taxes. That's $E.
- We add up any interest and/or revenue that the company earns from cash and securities. That's $R.
- Total
(called*Earnings before Interest and Taxes***EBIT**) = E - R dollars per year.
>Which explains why it's subracted from earnings, eh? Exactly. However, we'll use EBITDA (which is EBIT, excluding Depreciation and Amortization).
Now we generate a modified P/E Ratio (involving a modified
arnings)
EWe'll call it the M/E Ratio:
>Years? Them's the same dimensions as the ordinary, garden variety P/E ratio, eh?
>I assume you're going to modify the PEG Ratio ... again.
However, we'll also want a (pronounced gee-rank).:
gRANK
gRANK, the better.
>Why are you using Earnings Growth Rate? Why not use EBITDA Growth Rate? I would if I could, but where would I get that data? >I have no idea. Well, even if company debt and cash reserves etc. aren't available, one could always go back to , eh?
gNUM>Uh ... didn't I say I don't like that name? Pay attention. Notice that, for a high ranking (meaning a small >A small gRANK means a high rank? If a stock ranks #1 that's better than a rank of #10, eh? The smaller the better. Okay, for a high ranking (meaning a small ), we'd want small gRANK M/E and small Volatility and
large Earnings Growth Rate. Notice, too, that it contains lots of info about the company: past, present and future.
>Huh? Volatility (or Standard Deviation) is based upon past stock returns. (Volatility^{2} is the "Variance" of stock returns.)
M/E is a comment on the present state of the company.
Earnings Growth Rate is an estimate of future performance.
>Do you have an example?
>I'll take GE! Pay attention! We're not finished!
>So you're suggesting that MSFT, even with a larger P/E ratio, is the better buy? Me? Suggest? No, I just ... >Yeah, yeah. You just write tutorials ... and steal from a magazine! But doesn't this stuff make you think? >No! Note that the reciprocal, namely EBITDA / EV (measured in 1/years), can be regarded as income from your investment.
It'd be 1/17.4 = 0.057 or 5.7% per year for MSFT and 1/24.1 = 0.041 or 4.1% per year for GE. >And what about gRANK?
Wait'll I look up the Earnings Growth Rate and the Volatilities ... >I assume you have a spreadsheet that'll do all this. I'm working on it ... but I'm not sure I can find the pertinent data to download. Okay, I think I now have the numbers for Microsoft and General Electric ...
>Excuse me ... Where y'all going? >To buy some MSFT! Good luck ... |